You've been in for ages. Can I have a bat now?

A series solution about an ordinary point of a differential equation is always a Taylor series having a nonvanishing radius of convergence. A series solution about a singular point does not have this form (except in rare cases). Instead, it may be either a convergent series not in Taylor series form (such as a Frobenius series) or it may be a divergent series.

Let frantike Talbot triumph for a while, And like a Peacock ſweepe along his tayle, Wee’le pull his Plumes, and take away his Trayne, If Dolphin and the reſt will be but rul’d.

You sure took your time getting here!